C has a diameter with endpoints of (−5,3) and (1,−5). Select another point located on circle C

Answer 1

Hello Sky -

Thanks for your question. To begin, we need to figure out the coordinates of the center of the circle. Once we know the center, we can use the distance formula to find the radius of the circle, and then use the idea that all radii of the circle will be the same length in order to find other points that are separated from the center of the circle by a distance equal to the radius.

To find the coordinates of the center, we use the midpoint formula to find the midpoint between the endpoints of the diameter, since the center is halfway between the endpoints of the diameter:

x_midpoint = (x₁ + x₂)/2 = (-5 + 1)/2 = -2

y_midpoint = (y₁ + y₂)/2 = (3 + -5)/2 = -1

Thus, the coordinates of the center of this circle will be (-2, -1).

Now, we use the distance formula to figure out the radius of the circle, by calculating the distance between the center and one of the points on the circle (I will use (-5, 3), but either point will give the same result).

d = √((x₁ - x₂)² + (y₁ - y₂)²)

d = √((-2 - (-5))² + (-1 - 3)²)

d = √((3)² + (4)²)

d = √(9 + 16)

d = 5

Thus the radius of the circle is 5, and we can see that we can get from the center of the circle to our 2 existing points on the circle by going either left 3 units and up 4 units, or right 3 units and down 4 units, since both of these paths create right triangles with legs of 3 and 4 where the hypotenuse of 5 will be a radius of the circle.

Therefore, to get to another point on the circle, we can follow a similar path, just in a different direction: starting from the center, move 3 units horizontally and 4 units vertically, or 3 units vertically and 4 units horizontally, which will always mean that we wind up a total of 5 units from the center, and thus at a point on the circle.

If we go 3 units right from the center and 4 units up, that puts us at the point (5,3), which will be on the circle.

Another example would be to go 3 units left from the center and 4 units down, which would leave us at (-5, -5), another point on the circle.

Hope this helps! Let me know if any other questions and I will be happy to explain more.

Answer 2

You can find the center point ((-5+1)/2, (3-5)/2) = -2,-1) and the diameter is sqrt((-5-1)²+(3+5)²) = sqrt(6²+8²) = 10 and radius is 5)

The equation of the circle can be written as (x-k)² + (y-h)² = r² where (k,h) is the center and r the radius

(x+2)²+(y+1)² = 25

Pick any x that could be within the domain (safe is betwen -5 and 1)and you can solve for a y on the circle. x = -2 is easy, because (y+1)² = 25 y=4 so (-2,4)